Dynamo theory for the interface between the convection zone and the radiative interior of a star part i model equations and exact solutions

In this paper we derive a set of equations which model magnetic field generation and maintenance in a thin region, (- lo4) km thick, below the solar convection zone, and present some simple exact solutions. Energy to drive the dynamo is assumed to come from helical convection that overshoots into this region. Differential rotation and meridional circulation result only from feedbacks by the induced fields. The equations are derived for a homogeneous incompressible fluid in Cartesian geometry. The momentum equation, magnetic induction equation, and the continuity equation are included in the analysis. We assume velocity and magnetic field patterns have an aspect ratio of -1/10 (radial to horizontal scale), that the large scale velocities are smaller than the convection zone velocities, a few meters per sec, and the large scale magnetic fields are of the order of lo4 Gauss. Finally we assume that the time scale of interest is the advective time scale. Using these assumptions, we derive governing equations in which the Coriolis force balances the Lorentz force, the pressure gradient force, and the viscosity, and in which the magnetic fields are maintained by the ct effect but significantly modified by the above velocity fields. The results of this model will be discussed in following papers.